Precision Limits of Low-Energy GNSS Receivers Ken Pesyna and Todd Humphreys The University of Texas at Austin September 20, 2013

The GPS Dot Todd Humphreys Site: Search: “gps” 2 of 25

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Tradeoffs Size Weight Cost Precision Power 4

Tradeoffs Size Weight Cost Precision Power 5

Improvements Since 1990 Stagnation in battery energy density motivates the need for low power receivers: Batteries are not getting smaller 6

Simple Ways to Save Energy Modify parameters of interest: 1.Track fewer satellites, Nsv 2.Reduce the sampling rate, f s 3.Reduce integration time, T coh 4.Reduce quantization resolution, N bits 7

Assumptions 1.Consider only baseband processing 2.Baseband energy consumption is responsible for roughly half of the energy consumption in a GNSS chip [Tang 2012; Gramegna 2006] 3.Large assumed energy consumption by the signal correlation and accumulation operation (dot products) 8

Further Assumptions Assume that correlation and accumulation can be broken down into a series of bit-level additions. Each addition operation takes a fixed number of Joules, E A 9

Correlation and Accumulation Given K = f s *T coh samples to correlate K N-bit multiplies and K-1 (N+log 2 K)-bit adds o An N-bit add needs N 1-bit full adders o An N-bit multiply needs (N-1)*N 1-bit full adders Total number of 1-bit full adders N A in a CAA 10

Each 1-bit addition takes E A energy Energy consumed by a CAA, E CAA = E A ·N A Total energy consumed by all CAA operations Energy Required for Correlation and Accumulation 11

Given a fixed amount “E Total ” of Joules, what choices of T coh, f s, N, and N sv should we make to maximize positioning precision? Problem Statement 12

Given a fixed amount “E Total ” of Joules, what choices of T coh, f s, N, and N sv should we make to maximize positioning precision? Simplifications: Batch processing, a single fix No multipath (G w (f)= N 0 ) Optimal satellite geometry A bit of a fantasy world, but even still, we weren’t able to answer this question upfront Problem Statement 13

Step 1: Positioning Precision Positioning precision can be characterized by the RMS position-time error σ xyzt Geometric Dilution of Precision (GDOP) relates σ xyzt to the pseudorange error σ u : Optimal geometry: GDOP MIN = [Zhang 2009] 14

Step 2: Code Tracking Error Lower bound on coherent early-late discriminator design [Betz 09]: As the early-late spacing Δ  0, σ u,EL = σ u,CRLB 15

Step 2: Code Tracking Error Lower bound on code-tracking error σ u, independent of discriminator design [Betz 09]: Lower bound on coherent early-late discriminator design [Betz 09]: As the early-late spacing Δ  0, σ u,EL = σ u,CRLB, however, Δ min = 1/f s 16

Positioning Precision Step 2 17

Step 3: Effective Carrier-to-Noise Ratio The C s /N 0 is affected by the ADC quantization resolution N [Hegarty 2011] and others have shown that quantization resolution “N” decreases the overall signal power, C s The effective signal power at the output of the correlator C eff =C s /Lc 18 Function of N Function of G s (f) and f s

The C s /N 0 is affected by the ADC quantization resolution N [Hegarty 2011] and others have shown that quantization resolution “N” decreases the overall signal power, C s The effective signal power at the output of the correlator C eff =C s /Lc Step 3: Effective Carrier-to-Noise Ratio 19 Function of Quantization Precision “N”

Effective Carrier-to-Noise Ratio *Hegarty, ION GNSS 2010, Portland, OR 20

Quick Recap 4 parameters of interest: f s, N sv, T coh, N Derived baseband energy consumption: Derived lower bound on positioning precision: 21

Optimization Problem We have set up a constrained optimization problem to minimize σ xyzt for a given E Total 22

Tradeoff 1: Sampling Rate, f s vs Integration Time, T coh 23

Tradeoff 2: Sampling Rate, f s vs Quantization Resolution, N 24

Tradeoff 3: Number of SVs, N sv vs Integration Time, T coh 25

Optimization Solution versus Declining Energy 26

Conclusions 1.Investigated how certain parameters relate to energy consumption and positioning precision 2.Posed an optimization problem that solves for optimal values of the 4 parameters of interest under an energy constraint 3.Showed that the industry has come to anticipate many of the same conclusions 27

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